Having trouble remembering your multiplication tables? Here are just a few tricks to help.

I remember being in the first grade like it was yesterday. Besides having a major crush on a long-term substitute teacher and being the local savant on the 1960's space program, I was having the hardest time learning my multiplication tables. I could multiply numbers by 2, by 5, and by 10, but multiplying by 6 and 7 was difficult.

I've learned over the years that there are tricks and techniques to help. Here are just a few of them; I hope they'll help you remember your multiplication tables, too! The tricks I'm going to list work when multiplying together integers x and y between 1 and 10.

MULTIPLY BY 10: You just add zero on the end.

EXAMPLE: 3 times 10 is 30.

WHY IT WORKS: We work with digits, which are numbers written base 10. It probably comes as no surprise that the word "digit" refers both to a number and a finger; indeed, base 10 is what allows us to literally count on our fingers and toes! Multiplying by 10 is, by definition, the same as adding one more zero.

MULTIPLY BY 9: Say we want to multiply x times 9. You'll get a two-digit number whose sum is again 9, where the first digit of the number is one less than x.

EXAMPLE: We multiply 3 times 9. One less than 3 is 2, and 2 + 7 = 9, so 3 times 9 is 27.

WHY IT WORKS: One less than x is (x-1), and (x-1) + (10-x) = 9. Remember that placing a digit like (x-1) in the 10's place amounts to multiplying by 10. Performing a little algebra gives

10*(x-1) + (10-x) = 10*x - 10 + 10 - x = 10*x - x = 9*x.

(This is part of a phenomenon known as casting out nines. The idea is that a d-digit number with digits n1, n2, n3, ..., nd is divisible by 9 exactly when the sum of the digits (n1 + n2 + n3 + ... + nd) is divisible by 9.)

MULTIPLY NUMBERS BETWEEN 5 AND 10: I just learned about this trick earlier today by listening to Dennis Shasha on The Math Factor. Say that we want to multiply two positive integers x and y, where both integers are between 5 and 10. Perform the following steps:

1. On your left hand, count out x on your fingers. The number of fingers you should have pointing up is a = x-5, and the number of fingers you should have pointing down is 5-a.
2. On your right hand, count out y on your fingers. The number of fingers you should have pointing up is b = y-5, and the number of fingers you should have pointing down is 5-b .
3. Add together the number of fingers pointing up, namely a + b.
4. Multiply together the number of fingers pointing down, namely (5-a) * (5-b).
5. If you call the sum from (3) the first digit and the product from (4) the second digit, the resulting two digit number is the desired answer x*y.

EXAMPLE: Say we want to multiply 7 times 9. On one hand we place 2 fingers, and on the other we have 4 fingers. There are 2 fingers pointing up on the left hand and 4 fingers pointing up on the right hand; 2 plus 4 is 6. There are 3 fingers pointing down on the left hand and 1 finger pointing down on the right hand; 3 times 1 is 3. Hence 7 times 9 is 63.

WHY IT WORKS: On the left hand there are a = x-5 fingers pointing up and 5-a = 10-x fingers pointing down. Similarly, on the right hand there are a = y-5 fingers pointing up and 5-b = 10-y fingers pointing down. The total number of fingers pointing up is a+b = x+y-10, and the product of the fingers pointing down is (5-a)*(5-b) = (10-x)*(10-y). Remember that placing a digit like (a+b) in the 10's place amounts to multiplying by 10. Performing a little algebra gives

10*(a+b) + (5-a)*(5-b) = 10*x + 10*y - 100 + (10-x)*(10-y) = x*y